Theorem nnexpcl

Description: Closure of exponentiation of nonnegative integers. (Contributed by NM, 16-Dec-2005)

	Ref	Expression
Assertion	nnexpcl	⊢ ( ( 𝐴 ∈ ℕ ∧ 𝑁 ∈ ℕ0 ) → ( 𝐴 ↑ 𝑁 ) ∈ ℕ )

Proof

Step	Hyp	Ref	Expression
1		nnsscn	⊢ ℕ ⊆ ℂ
2		nnmulcl	⊢ ( ( 𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ ) → ( 𝑥 · 𝑦 ) ∈ ℕ )
3		1nn	⊢ 1 ∈ ℕ
4	1 2 3	expcllem	⊢ ( ( 𝐴 ∈ ℕ ∧ 𝑁 ∈ ℕ0 ) → ( 𝐴 ↑ 𝑁 ) ∈ ℕ )